The 31-year-old midfielder has been a key player for the Cottagers so far this season, scoring five goals in 20 Premier League appearances.
Fulham are in a fight for top-flight survival, with Saturday's 2-0 defeat to Arsenal leaving them 17th, just one point better off than the bottom three.
Sidwell has been a shining light in a disappointing campaign but is out of contract at the end of this season, having rejected a 12-month extension back in October.
Meulensteen has urged the club's hierarchy to find a 'suitable solution' to the contract stand-off, and told the Fulham Chronicle: "It's clear that Steve would love to stay here.
"He's doing very well here so I hope we come to a suitable solution.
"He will definitely get the credit he deserves from us. His fellow team-mates respect him, as do the staff, for the tremendous job he does for us."